add panel to invoke_props_dialog
Import the classes, use their draw method.

If you enable developer tools in preferences you can right click in the UI and choose view source.  Doing this for the cycles render panels you will see the classes are defined in ui.pyof the cycles addon.  Can import them with 

from cycles.ui import CYCLES_RENDER_PT_sampling, CYCLES_RENDER_PT_sampling_advanced


Note: some areas have their own context members.  Eg PROPERTIES might use context.speaker for the data panel of a speaker object. This may lead to errors using the method outlined here.


Test run in text editor

Test code.

import bpy
from cycles.ui import CYCLES_RENDER_PT_sampling, CYCLES_RENDER_PT_sampling_advanced

class TEST_OT_popup(bpy.types.Operator):
    bl_idname = 'test.popup'
    bl_description = 'test'
    bl_label = 'test'
    bl_options = {'UNDO'}

    @classmethod
    def poll(cls, context): 
        return CYCLES_RENDER_PT_sampling.poll(context)

    def invoke(self, context, event):
        return context.window_manager.invoke_props_dialog(self, width=250)

    def execute(self, context):
        return {'FINISHED'}

    def draw(self, context):
        layout = self.layout

        CYCLES_RENDER_PT_sampling.draw(self, context)
        layout.label(text="Advanced")
        CYCLES_RENDER_PT_sampling_advanced.draw(self, context)

# for testing      
def register():
    bpy.utils.register_class(TEST_OT_popup)

register()
bpy.ops.test.popup('INVOKE_DEFAULT')